An improved braking cable-payout mechanism has been developed. Whereas other such mechanisms operate at payout speeds that vary with the length of payout, this mechanism operates at approximately constant payout speed, regardless of the length of cord that has already been paid out. The present mechanism includes a spool, a capstan assembly, and centrifugal brakes. The spool is used to store the cord and, unlike in the prior mechanism, is not involved in the primary braking function. That is, the spool operates in such a way that the cord is unwound from the spool at low tension. The spool is connected to the rest of the mechanism through a constant- torque slip clutch. The clutch must slip in order to pay out the cord. As the cord leaves the spool, it passes into the capstan assembly, wherein its direction is changed by use of the first of three idler sheaves and it is then routed into the first of three grooves on a capstan. After completing less than a full circle in the first groove, the cord passes over the second idler sheave, which is positioned to enable the cord to make the transition to the second groove on the capstan. Similarly, a third idler sheave enables the cord to make the transition to the third groove on the capstan. After traveling less than a full circle in the third groove, the cord leaves the capstan along the payout path. The total wrap angle afforded by this capstan-and-idler arrangement is large enough to prevent slippage between the cord and the capstan. The capstan is connected to a shaft that, in turn, is connected to a centrifugal brake. Hence, the effective payout radius, for purposes of braking, is not the varying radius of the remaining cord on the spool but, rather, the constant radius of the grooves in the capstan. The payout speed is determined primarily by this radius and by the characteristics of the centrifugal brake. Therefore, the payout speed is more nearly constant in this mechanism than in the prior mechanism.